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Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
To do:
Here, we have to write the decimal expansions of the given rational in numbers Question 1.
Solution:
(i) $\frac{13}{3125}=\frac{13}{5^5}$
Multiply the numerator and denominator by $2^5$ so that the denominator becomes a multiple of $10^r$, where r is any positive integer.
Therefore,
$\frac{13}{3125}=\frac{13}{5^5}$
$=\frac{13\times2^5}{5^5\times2^5}$
$=\frac{13\times32}{(2\times5)^5}$
$=\frac{416}{10^5}$
$=\frac{416}{100000}$
$=0.00416$
The decimal expansion of the given rational number is $0.00416$.
(ii) $\frac{17}{8}=\frac{17}{2^3}$
Multiply the numerator and denominator by $5^3$ so that the denominator becomes a multiple of $10^r$, where r is any positive integer.
Therefore,
$\frac{17}{8}=\frac{17}{2^3}$
$=\frac{17\times5^3}{2^3\times5^3}$
$=\frac{17\times125}{(2\times5)^3}$
$=\frac{2125}{10^3}$
$=\frac{2125}{1000}$
$=2.125$
The decimal expansion of the given rational number is $2.125$.
(iii) $\frac{15}{1600}=\frac{15}{2^6\times5^2}$
Multiply the numerator and denominator by $5^4$ so that the denominator becomes a multiple of $10^r$, where r is any positive integer.
Therefore,
$\frac{15}{1600}=\frac{15}{2^6\times5^2}$
$=\frac{15\times5^4}{2^6\times5^2\times5^4}$
$=\frac{15\times625}{(2\times5)^6}$
$=\frac{9375}{10^6}$
$=\frac{9375}{1000000}$
$=0.009375$
The decimal expansion of the given rational number is $0.009375$.
(iv) $\frac{23}{2^{3} \times 5^{2}}$
Multiply the numerator and denominator by $5^1$ so that the denominator becomes a multiple of $10^r$, where r is any positive integer.
Therefore,
$\frac{23}{2^{3} \times 5^{2}}=\frac{23\times5^1}{2^3\times5^2\times5^1}$
$=\frac{23\times5}{(2\times5)^3}$
$=\frac{115}{10^3}$
$=\frac{115}{1000}$
$=0.115$
The decimal expansion of the given rational number is $0.115$.
(v) $\frac{6}{15}=\frac{2}{5}$
Multiply the numerator and denominator by $2^1$ so that the denominator becomes a multiple of $10^r$, where r is any positive integer.
Therefore,
$\frac{6}{15}=\frac{2}{5}$
$=\frac{2\times2^1}{(5\times2)^1}$
$=\frac{4}{10^1}$
$=\frac{4}{10}$
$=0.4$
The decimal expansion of the given rational number is $0.4$.
(vi) $\frac{35}{50}=\frac{5\times7}{5\times10}$
$=\frac{7}{10}$
$=0.7$
The decimal expansion of the given rational number is $0.7$.
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